The mistakes children make in mathematics are usually not just ‘mistakes’ – they are often intelligent generalizations from previous learning. Following several decades of academic study of such mistakes, the phrase ‘errors and misconceptions’ has recently entered the vocabulary of mathematics teacher education and has become prominent in the curriculum for initial teacher education.

The popular view of children’s errors and misconceptions is that they should be corrected as soon as possible. The authors contest this, perceiving them as potential windows into children’s mathematics. Errors may diagnose significant ways of thinking and stages in learning that highlight important opportunities for new learning.

This book uses extensive, original data from the authors’ own research on children’s performance, errors and misconceptions across the mathematics curriculum. It progressively develops concepts for teachers to use in organizing their understanding and knowledge of children’s mathematics, offers practical guidance for classroom teaching and concludes with theoretical accounts of learning and teaching.

Children’s Mathematics 4-15 is a groundbreaking book, which transforms research on diagnostic errors into knowledge for teaching, teacher education and research on teaching. It is essential reading for teachers, students on undergraduate teacher training courses and graduate and PGCE mathematics teacher trainees, as well as teacher educators and researchers.

Anyway, I found an online version of the text via Google Books, which is embedded below. However, pages 82, 83, 87, 88, 94, 95, 101 and 102 are not included in the preview due to copyright reasons. Although the content that is there gives a reasonable account of the subject and includes many examples of good practice. Clearly, the lack of the full chapter really doesn’t help!

Having just finished my work with a bunch of Year 3 – who are far fussier than I remember – I feel quite pleased.

They were able to build and complete the secret construction quicker and more accurately than I first hoped, they used a good range of language – next to, on top of, it looks like a house (!) – and generally they worked well together.

I used magnetic polydron to build the shape for the secret construction again. In fact, I decided I would use the same shape entirely for the job, with the same colours too! It was clear that they had done this activity before. They knew what was expected of them – although that didn’t stop me from telling them – and how to go about getting to a satisfactory end. Certainly the secret construction is an activity that works lower down school than I am used to and I can see it being of value to them from a vocabulary, communication and shape point of view. The only problem with magnetic polydron is that currently we only have squares and equilateral triangles, which limits the number of shapes we can make. I have included some of their creativity with the equipment from the very end of the session when I let them have a little play…

During the string activity, it was clear that they had limited knowledge of shape names. They struggled to predict what shapes would be made – although a couple did correctly identify a hexagon. When a star shape was created with a heptagon inside, I already knew they wouldn’t have a clue what it was, so focussed on the outside shapes instead.

Now, because of troubles when lowering the string to the floor, we ended up with some unusual patterns around the edge. We had trapeziums and triangles… the trapezium intruiged me. So I asked them what they thought it was called. Instantly, we had the name ‘square‘, which lead me to ask why they considered it a square. The response I got showed their knowledge, but also immediately reminded them they were wrong. They said, “It has four sides… but they’re not the same size, so it can’t be a square.” Intelligent thinking! This lead another child to say, “Well it must be a rectangle then!” Prompting another to say, “But it doesn’t have four right angles.”

While I wasn’t expecting this at all, it showed that a simple thing can generate such a wonderful discussion. To me it doesn’t matter that they didn’t know what a trapezium was, it was valuable enough for me to go back to their teacher and tell him that those children knew the properties, roughly anyway, of a rectangle and a square. And thinking about it, that’s all they should know. After all, it’s the fourth week of their first half term in Key Stage 2 – their knowledge of shape hasn’t been touched since the back end of Year 2 anyway, and that would consist of looking at the names of basic shapes.

If it taught me one thing, it’s that I haven’t been around Year 3 enough lately, that I’ve become used to the language and abilities of Years 5 and 6 so much that I’ve desensitised myself from children further down the school.

I certainly need to make time to work with them more throughout this course.

The NCETM has a series of tools for analysing how confident you feel about various areas of mathematics. Part of the MaST programme requires me to complete each area over time. I also have to complete the sections for a range of Key Stages – 1, 2 and 3 – to demonstrate a broad knowledge of the subject.

Here are my results for the Understanding Shape/Geometry sections. (1 is not confident and 4 is very confident)

How confident are you that you understand the relationship between angle as a measure of turn?

How confident are you that you can give relevant examples to illustrate the meaning of reflection?

How confident are you that you can give relevant examples to illustrate the meaning of line or reflection symmetry?

How confident are you that you know common side, angle and symmetry properties of polygons?

How confident are you that you know common side, angle and symmetry properties of triangles?

How confident are you that you know common side, angle and symmetry properties of squares and rectangles?

I answered 4 for each of these, giving me an outcome of very confident. I chose 4 for each of the answers as, reading through the examples given, I use the techniques described and go deeper too, being a Key Stage 2 teacher.

How confident are you that you can show that the sum of the angles in a triangle is 180° in two different ways?

Again, I answered 4 for each of these questions, giving me an outcome of very confident. I chose 4 because of the ways I have used to teach shape over the years in Years 5 & 6. I use pull up nets to show how the 5 Platonic solids are made, regularly discuss the properties of shapes – especially the range of triangles – with my class. One minor concern was the use of ICT in the first 6 questions, but I consider my SMART Notebook slides to be using ICT and I rarely teach a maths lesson without one.

How confident are you that you are aware of a range of visualisation activities to help pupils to appreciate properties and transformations of shapes? (3)

How confident are you that you understand through practical activity and the use of ICT the meaning of translation? (4)

How confident are you that you understand through practical activity and the use of ICT the meaning of reflection? (4)

How confident are you that you understand through practical activity and the use of ICT the meaning of rotation? (4)

How confident are you that you understand through practical activity and the use of ICT the meaning of enlargement? (4)

How confident are you that you know the meanings of alternate angles, corresponding angles, supplementary angles, complementary angles? (3)

How confident are you that you can prove that the exterior angle of a triangle is equal to the sum of the two interior opposite angles, the sum of the angles in a triangle is 180° and the sum of the exterior angles of any polygon is 360°? (4)

How confident are you that you can prove that the opposite angles of a parallelogram are equal? (3)

How confident are you that you know the conditions for congruent triangles and can prove that the base angles of an isosceles triangle are equal? (3)

How confident are you that you know how to establish through geometrical reasoning the side, angle and diagonal properties of quadrilaterals? (3)

How confident are you that you know how to execute and prove the standard straight−edge and compass constructions? (3)

How confident are you that you know how to describe simple loci? (2)

How confident are you that you know how to explain and prove some circle theorems? (3)

How confident are you that you understand Pythagoras’ theorem and its application to solving mathematical problems? (4)

How confident are you that you can explain the conditions for similar triangles? (4)

This held some trepidation for me, as I haven’t ever really considered the Key Stage 3 curriculum before now for geometry while teaching. My answers are in brackets above and a mainly a mix of 3s and 4s with one 2. This gave me and outcome of confident. The 2 is for the question about simple loci – a choice made because I can’t remember having done any locus work in years! (The locus of a point is its path when it moves according to given rules or conditions. The plural is loci.) I think, having read the examples on the NCETM site, that I could certainly do the work for myself, but would probably struggle to teach it.

Where I chose 3, it is often because I felt I fully understood most of the content but there were areas where I may not have been able to give examples. In question 6, for instance, I would be fine with alternate angles, corresponding angles and complementary angles but may confuse supplementary angles.

Clearly from this, I need to develop my knowledge of some of the Key Stage 3 geometry material.